CROSSREFERENCE TO RELATED APPLICATIONS

[0001]
This application claims priority to U.S. patent application Ser. No. 60/368,212, filed Mar. 19, 2002, titled “Non Linear Technique for Large Angle MicroMirror Control”, the entirety of which provisional application is incorporated by reference herein.
FIELD OF THE INVENTION

[0002]
The invention relates generally to electrostatic mirror arrays used in optical switches and optical networks. More particularly, the invention relates to a method of stabilizing a position control system for an electrostatic mirror.
BACKGROUND

[0003]
Electrostatic micromirror arrays are becoming more attractive for use in optical communication networks. For example, they can be used in optical switching to actively route optical signals between input and output channels. The overall performance of the optical switch is determined, in part, by the performance of the micromirror arrays. At large deflection angles, an array mirror can be unstable, that is, the mirrors cannot be accurately maintained at the desired angle of deflection.

[0004]
Mirror instabilities are typically reduced using a sensor based feedback system. For twoaxis systems, substantial interactions between the rotation axes often occur. Consequently, when using a linear or quasilinear control system, instabilities are apparent if the mirror is deflected through a large angle on each axis. Thus the mirror is restricted to applications requiring only a limited range of rotation. For systems utilizing such mirrors, other system design parameters can be adjusted to accommodate the limited range. Unfortunately, the result is often a larger package size or reduced system performance.
SUMMARY

[0005]
In one aspect, the invention features a method for controlling the angular position of a mirror. The angular position of the mirror is sensed and a first linear control signal is generated in response to the angular position. A first nonlinear control signal is generated to control the angular position of the mirror. The first nonlinear control signal is responsive to the linear control signal.

[0006]
In another aspect, the invention features a system for controlling the angular position of a mirror. The system includes a linear control module for generating a linear control signal in response to the angular position. The system also includes a nonlinear mapping module that converts the linear control signal into a nonlinear control signal for angularly positioning the mirror. In one embodiment, the system includes a position sensor for determining the angular position of the mirror about at least one axis. In another embodiment, the system includes a coefficient adaptation module in communication with the nonlinear mapping module.

[0007]
In another aspect, the invention features a method for adapting a nonlinear mirror control system for a mirror. A linear control value is determined for each of three angular positions. Quadratic coefficients are determined in response to the three angular positions and the three linear control values. An extrapolated linear control value based on the quadratic coefficient is determined and an adapted coefficient is calculated in response to the extrapolated linear control value.
BRIEF DESCRIPTION OF THE DRAWINGS

[0008]
The above and further advantages of this invention may be better understood by referring to the following description in conjunction with the accompanying drawings, in which like numerals indicate like structural elements and features in various figures. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention.

[0009]
[0009]FIG. 1 is an illustration of a prior art micromirror having two axes of rotation.

[0010]
[0010]FIG. 2 is a plot of electrostatic torque as a function of rotation angle and an applied control voltage for a two axis mirror with the angular position defined about one axis held constant.

[0011]
[0011]FIG. 3 is a block diagram of an embodiment of a mirror control system used to control one axis according to the present invention.

[0012]
[0012]FIG. 4 is a flowchart representation of an embodiment of a method for controlling the angular position of a mirror according to the present invention.

[0013]
[0013]FIG. 5 is a plot of the linear control value 4 as a function of normalized angle before and after application of an embodiment of an adaptation method of the present invention.

[0014]
[0014]FIG. 6 is a plot of the angular positioning of the mirror during an embodiment of the adaptation process of the present invention.

[0015]
[0015]FIGS. 7A and 7B are plots of the values of k_{x}, k_{y }and k_{xy }during an embodiment of an adaptation process of the present invention.
DETAILED DESCRIPTION

[0016]
Various feedback systems are used to control twoaxis electrostatic mirrors. These feedback systems, however, are unable to eliminate mirror instabilities at large deflection angles due to interactions, or coupling, between the two axes of rotation. The present invention relates to a method and system for controlling mirrors that reduces or substantially eliminates this instability.

[0017]
In brief overview, the present invention provides for controlling the angular position of a mirror. Because of the inherent nonlinearities exhibited by the mirror at large deflection angles, linear control methods limit the useful angular range of operation. According to the present invention, a control parameter which linearizes a control system is determined. Conventional control modules implementing linear control methods, such as proportionalintegralderivative (PID) control or state space control are implemented with the control parameter. However, instead of directly controlling the mirror with the output signal provided by the linear control module, the output signal is used to determine a nonlinear control signal. The nonlinear control signal is then applied directly to the mirror system to control the angular position of the mirror. As a result, the mirror can be operated over a wider range of deflection angles.

[0018]
Referring to FIG. 1, an electrostatic twoaxis (i.e., 2D) scan mirror 10 includes an inner pair of hinges 14 supported by an inner frame 20 that permit rotation of a reflective surface 16 about the xaxis 18. The mirror 10 also includes an outer pair of hinges 22 supported by outer frame 24 for rotation of the reflective surface 16 and inner frame 20 about the yaxis 26. The mirror 10 represents one of a plurality of mirrors in a micromirror array and lies adjacent to and parallel to a set of electrode plates (not shown) on an underlying substrate. For each mirror 10, there are typically four electrode plates arranged in quadrants according to the projection of the xaxis 18 and yaxis 26 onto the plane of the electrode plates.

[0019]
The angular position of the mirror 10 is determined according to the differences between the voltages of the electrode plates. For example, to position the mirror 10 at a new angle about the xaxis 18, the voltage applied to the two electrodes on one side of the xaxis 18 is increased above a bias voltage by a fixed value and the voltage applied to the electrodes on the other side of the xaxis 18 is decreased by approximately the same value. Consequently, the spatial variation in the electric field established between the electrodes and the mirror 10 causes the mirror 10 to rotate about the xaxis 18 to the new position. Similarly, to rotate the mirror 10 about the yaxis 26, the voltage applied to the electrodes on one side of the yaxis 26 is increased by a certain value and the voltage applied to the electrodes on the opposite side of the yaxis is decreased by the same value.

[0020]
The dynamics of the electrostatic mirror
10 are described by the following equations:
$\begin{array}{cc}\frac{\uf74c{x}_{1}}{\uf74ct}={x}_{2}& \left(1\ue89ea\right)\\ \frac{\uf74c{x}_{2}}{\uf74ct}=\frac{{k}_{\mathrm{vx}}}{{J}_{x}}\xb7{x}_{2}\frac{{k}_{\mathrm{sx}}}{{J}_{x}}\xb7{x}_{1}+\frac{{f}_{x}\ue8a0\left({x}_{1},\text{\hspace{1em}}\ue89e{y}_{1},\text{\hspace{1em}}\ue89e{u}_{x}\right)}{{J}_{x}}& \left(2\ue89e\text{\hspace{1em}}\ue89eb\right)\\ \frac{\uf74c{y}_{1}}{\uf74ct}={y}_{2}& \left(1\ue89ec\right)\\ \frac{\uf74c{y}_{2}}{\uf74ct}=\frac{{k}_{\mathrm{vy}}}{{J}_{y}}\xb7{y}_{2}\frac{{k}_{\mathrm{sy}}}{{J}_{y}}\xb7{y}_{1}+\frac{{f}_{y}({y}_{1},\text{\hspace{1em}}\ue89e{u}_{y)}}{{J}_{y}}& \left(1\ue89ed\right)\end{array}$

[0021]
in which x_{1 }and x_{2 }are the xaxis radian position and xaxis angular velocity, respectively, and y_{1 }and y_{2 }are the yaxis radian position and yaxis angular velocity, respectively. J_{x }and J_{y }represent the mirror moments of inertia about the xaxis 18 and yaxis 26, respectively, and k_{vx}, k_{sx}, k_{vy }and k_{sy }represent the velocity and position force constants for the xaxis 18 and yaxis 26, respectively, u_{x }and u_{y }are the control voltage signals applied to the electrodes to position the mirror 10 about the xaxis 18 and yaxis 26, respectively. The function f_{x}(x_{1}, y_{1}, u_{x}) represents the electrostatic torque for the mirror 10 about the xaxis 18 and includes the coupling effect from the yaxis 26 to the xaxis 18. The function f_{y}(y_{1}, u_{y}) represents the electrostatic torque for the mirror 10 about the yaxis 26 and does not include any coupling effect from the xaxis 18 because this interaction is generally small and therefore is ignored.

[0022]
[0022]FIG. 2 shows an example of the mirror torque as a function of the deflection angle θ and the control voltage u for the electrostatic mirror 10. In this example, the mirror 10 is fixed such that no rotation is possible about the yaxis 26.

[0023]
The angle θ_{x }axis represents the angular position of the mirror 10 about the xaxis 18 in degrees and the voltage axis represents the voltage of the control signal u_{x }used to drive the mirror in rotation about the xaxis 18. The torque axis represents the torque applied to the mirror 10 to maintain its angular position θ_{x}. As evident by the electrostatic torque “surface” 28, the torque increases rapidly in a nonlinear manner beyond approximately ±2°.

[0024]
Prior art control methods implementing linear systems with constant gains are stable only in the region in which the torque relationship is substantially planar. Consequently, for the illustrated relationship, the range of rotation for each axis for such prior art control systems is typically limited to approximately ±2°.

[0025]
The response of the mirror 10 to the nonlinear control signal u_{x }(or u_{y}) depends in part on the separation of the mirror 10 from its driving electrodes and the residual tilt angle defined between the mirror 10 and the underlying substrate. For micromirror arrays, structural variations in the individual mirrors resulting from the fabrication process are common, therefore, it is desirable to adapt, or tune, each mirror independently. The adaptation method of the present invention accounts for the variances in each mirror to thereby extend the rotation range of each mirror beyond what is generally achievable by treating all the mirrors identically. Moreover, because the response of each mirror is typically dependent on the polarity (i.e., sign) of the rotation about each axis, it is advantageous to apply the adaptation method to each angular quadrant of operation for each mirror.

[0026]
For simplicity, the nonlinear control system and method described below are generally described with respect to the xaxis 18. It should be understood by those of ordinary skill that the principles of the present invention also apply to rotation about the yaxis 26.

[0027]
Referring again to FIG. 1, the mirror
10 is hinged along its centerline and rotates in angle θ
_{x }about the xaxis
18. As previously described, the mirror
10 has two pairs of electrode plates driven in opposition. The mirror torque is given approximately by
$\begin{array}{cc}f\ue8a0\left({x}_{1},\text{\hspace{1em}}\ue89e{y}_{1},\text{\hspace{1em}}\ue89eu\right)\equiv T=\mathrm{Kt}\xb7\left[\frac{{\left(V+u\right)}^{2}}{{\left({d}_{o}{L}_{x}\xb7x{L}_{y}\xb7x\xb7y\right)}^{2}}\frac{{\left(Vu\right)}^{2}}{{{d}_{o}+{L}_{x}\xb7x{L}_{y}\xb7\leftx\xb7y\right)}^{2}}\right]& \left(2\right)\end{array}$

[0028]
in which T is the torque on the mirror
10, Kt is a proportionality constant, V is the mirror bias voltage and u is the control signal voltage which is applied in opposite polarity to the pairs of electrode plates. Variables x and y are the angular positions in radians about the xaxis
18 and yaxis
26, respectively, d
_{0 }is the gap, or separation, between the electrodes and the mirror
10 when the voltage control signal u is zero. L
_{x }and L
_{y }are the influence coefficients that describe how the gap d
_{0 }varies when the mirror
10 is rotated about the xaxis
18 and yaxis
26, respectively. In the illustrated embodiment, the influence coefficients L
_{x }and L
_{y }are approximately 2d
_{0}/L and 2d
_{0}/W, respectively where L is the length of the mirror
10 along the yaxis
26 and W is the length of the mirror
10 along the xaxis
18. The mirror system dynamics are described by the following nonlinear equation using equations (1a through 1d) and equation (2):
$\begin{array}{cc}{J}_{x}\xb7\ddot{x}+{k}_{\mathrm{vx}}\xb7\stackrel{.}{x}+{k}_{\mathrm{sx}}\xb7x=\mathrm{Kt}\xb7\left[\frac{{\left(V+u\right)}^{2}}{{\left({d}_{o}{L}_{e}\xb7x{L}_{y}\xb7\leftx\xb7y\right\right)}^{2}}\frac{{\left(\stackrel{.}{V}u\right)}^{2}}{{\left({d}_{o}+{L}_{e}\xb7x{L}_{y}\xb7\leftx\xb7y\right\right)}^{2}}\right]& \left(3\right)\end{array}$

[0029]
in which {umlaut over (x)} and {dot over (x)} are the angular acceleration and angular velocity, respectively, about the xaxis
18. The mirror system described by equation (3) is “inputlinearized” using the following substitution:
$\begin{array}{cc}{k}_{g}\xb7\xi \equiv f\ue8a0\left({x}_{1},\text{\hspace{1em}}\ue89e{y}_{1},\text{\hspace{1em}}\ue89eu\right)\xb7\frac{{d}_{o}^{2}}{{K}_{t}}& \left(4\right)\end{array}$

[0030]
in which ξ is a linear control variable in the linearized input space that is proportional to the computed torque and k
_{g }is a proportional gain constant. By substitution, equation (3) can now be expressed as a linear equation in the linearized input space as:
$\begin{array}{cc}{J}_{x}\xb7\ddot{x}+{k}_{\mathrm{vx}}\xb7\stackrel{.}{x}+{k}_{\mathrm{sx}}\xb7{k}_{g}\xb7\xi \xb7\frac{{K}_{t}}{{d}_{o}^{2}}& \left(5\right)\end{array}$

[0031]
Thus, the mapping of the control variable ξ to the angular position x (or y) of the mirror 10 is linear. The linear control variable ξ is calculated using any linear systems technique. For example, the linear control module (i.e., the system controller) can implement methods such as PID, state estimation or discrete sliding mode control to generate the linear control variable ξ. In an exemplary embodiment using PID, the linear control module receives an error signal ε that represents the difference between a target angle value and an actual sensed angle value x (or y). Thus, ξ includes a component that is proportional to the error signal ε, a component that is proportional to the derivative of the error signal ε and a component that is proportional to the integral of the error signal ε.

[0032]
The output control voltage u provided to the system for stable operation is determined from the following normalized equation derived from equations (3) and (5)
$\begin{array}{cc}\xi =\left[\frac{{\left(V+u\right)}^{2}}{{\left(1{k}_{x}\xb7x{k}_{\mathrm{xy}}\xb7\leftx\xb7y\right\right)}^{2}}\frac{{\left(Vu\right)}^{2}}{{\left(1+{k}_{x}\xb7x{k}_{\mathrm{xy}}\xb7\leftx\xb7y\right\right)}^{2}}\right]& \left(6\right)\end{array}$

[0033]
in which the coefficients k_{x }and k_{xy }are the influence coefficients L_{e }and L_{y }described above divided by the gap d_{0}. The control voltage u is calculated each time the mirror position is determined (i.e., sampled) and used to maintain the desired angular position until the next sampling is completed.

[0034]
[0034]FIG. 3 is a block diagram of a mirror control system 30 for controlling positioning of the mirror 10 about the xaxis 18 (FIG. 1). The control system 30 includes a pair of angular position sensors 34 and 38 in communication with a pair of analogtodigital converters (ADCs) 42, 46, respectively, and a differencing element 50 in communication with the xaxis ADC 42.

[0035]
The control system 30 also includes a linear control module 54 in communication with the differencing element 50, a nonlinear mapping module 58 in communication with the ADCs 42, 46 and the linear control module 54, an adaptation module 62 in communication with the nonlinear mapping module 58, and a digitaltoanalog converter (DAC) 66 in communication with the nonlinear mapping module 58 and the mirror 10′. A flowchart representation for a method of controlling the angular position of the mirror 10′ using the control system 30 of FIG. 3 is shown in FIG. 4.

[0036]
Referring to both FIGS. 3 and 4, the pair of angular sensors (e.g., piezoresistive sensors) 34, 38 are used to sense (step 70) the angular positions of the mirror 10 about the xaxis 18 and yaxis 26, respectively. The sensors 34, 38 generate analog signals APOSX and APOSY having a value indicative of the current angular positions x and y of the mirror 10. The analog signals APOSX, APOSY are converted to digital signals DPOSX and DPOSY by ADCs 42 and 46, respectively. A difference signal ε, representative of the difference between the digital position DPOSX for the xaxis 18 and a reference signal XREF indicating the desired angular position (i.e., reference position), is generated (step 74) by the differencing element 50.

[0037]
The linear control module 34 generates (step 78) a linear control signal 4 having a value responsive to the difference signal ε. The nonlinear mapping module 38 then determines (step 82) the appropriate value of the nonlinear control signal u according to equation (6). In one embodiment, the nonlinear mapping module 38 is a digital signal processor (DSP). Coefficients k_{x }and k_{y }corresponding to the given mirror 10 are provided by the adaptation module 62 to the nonlinear mapping module 58. The function of the adaptation module 62 is described in more detail below. The nonlinear control signal u is converted to an analog voltage signal by the DAC 66 and applied (i.e., added or subtracted) (step 86) to the bias voltage of the electrodes.

[0038]
This process of sensing angular position, generating the difference signal ε, generating the linear control signal ξ and determining the nonlinear control signal u is repeated each time the mirror angular position is sampled. The sampling frequency is typically determined according to the specific requirements of the application employing the mirror system.

[0039]
In one embodiment, the control voltage u is determined for each sample period by calculating the following variables:

g _{1}=(1−k _{x} ·x−k _{xy} ·x·y)^{2 }

g _{2}=−(1+k _{x} ·x−k _{xy} ·x·y)^{2 }

c _{0} =V ^{2}·(g _{1} +g _{2})−ξ·g _{1} ·g _{2} (7a)

c _{1}=2·V(g _{1} +g _{2})

c _{2}=(g _{1} +g _{2})

[0040]
The control voltage u is then determined by solving the following equation:

c _{0} +c _{1} ·u+c _{2} ·u ^{2}=0 (7b)

[0041]
As previously described, the performance of individual mirrors 10 in a micromirror array can differ due to variations in the mirror structures. For example, the separation d_{0 }of each mirror 10 from the substrate typically varies slightly between mirrors, and particularly between micromirror arrays. Furthermore, the coefficients k_{x}, k_{xy }and k_{y }vary at large angles because the actual gap between the mirror 10 and the substrate becomes small compared to d_{0}. In addition, any offset tilt for mirrors at rest and/or variations in the sensitivity of the angular position sensors 34, 38 affect k_{x}, k_{xy }and k_{y}. If not addressed, these variations have a significant impact on the stability of the mirror system 30 and, therefore, the control parameters. In order to expand the useful range of angular motion, the coefficients k_{x}, k_{xy }and k_{y }are adaptively determined.

[0042]
Referring to FIG. 5, the control signal ξ is plotted as a function (dashed line 90) that is linear with angle x. However, due to the variations in the mirror structures, the functional relationship between the control signal ξ and the angular position x for an individual mirror 10 in a micromirror array is more generally described by a curve that turns either upward (e.g., solid line 94) or downward as x increases. In accordance with the present invention, a calibration, or adaptation procedure, is implemented wherein the coefficients k_{x }and k_{xy }are tuned, or adapted, for each mirror 10 so that the control signal ξ is substantially linear with angle x.

[0043]
The adaptation procedure for the coefficient k_{x }includes determining the initial control signal ξ for each of three angular positions: x=0, x=θ_{1 }and x≈θ_{2 }as shown by line 98 in FIG. 6. The resulting three points are fit to a quadratic equation

a _{0} a _{1} ·x+a _{2} ·x ^{2}=ξ (8)

[0044]
Because a linear dependence is expected, the coefficients a
_{0 }and a
_{1 }resulting from the fit to equation (8) are then used to extrapolate the value of ξ at x=θ
_{2 }from the linear relationship ξ
_{0}=a
_{0}+a
_{1 }θ
_{2}. While the mirror is rotated to a position at x=θ
_{2}, the coefficient k
_{x }is adapted as follows:
$\begin{array}{cc}{k}_{x}\ue8a0\left(i+1\right)={k}_{x}\ue8a0\left(i\right)+\sigma \xb7\frac{\left({\xi}_{0}\xi \right)}{\left{\xi}_{0}\right}& \left(9\right)\end{array}$

[0045]
in which σ is the adaptation parameter used in the relaxation. Thus, as the mirror is driven from x=0 to x=θ_{2}, the value of the coefficient k_{x }(i.e., k_{x}(i+1)) is repeatedly calculated from a previously calculated value (i.e., k_{x}(i)) and the current control parameter value ξ using equation (9). The adaptation of k_{x }continues until the relationship between the control parameter and angle x is sufficiently linear. In one embodiment, the adaptation parameter σ is a fixed design constant and the calculation is repeated until a predetermined number of iterations is reached. In another embodiment, calculations continue until the difference k_{x}(i+1)−k_{x}(i) is less than a predetermined error value. The coefficient k_{y }is adapted in the same manner as the adaptation of the coefficient k_{x}. An example adaptation of coefficient k_{x }and coefficient k_{y }is shown in FIG. 7A in which the horizontal axis represents the iteration number (i) of the calculation of coefficient k_{x }and the vertical axis is the normalized value of k_{x }or k_{y}.

[0046]
To adapt the coefficient k
_{xy}, the value of the control parameter ξ is determined at (x=θ
_{2}, y=0). The mirror is then positioned at (x=θ
_{2}, y=θ
_{2}) (See line
102 of FIG. 6) If there is no coupling, the value of control parameter ξ remains at ξ
_{0}. While the mirror is rotated to (x=θ
_{2}, y=θ
_{2}) the coefficient k
_{xy }is adapted as follows:
$\begin{array}{cc}{k}_{\mathrm{xy}}\ue8a0\left(i+1\right)={k}_{\mathrm{xy}}\ue8a0\left(i\right)+\sigma \xb7\frac{\left({\xi}_{0}\xi \right)}{\left{\xi}_{0}\right}& \left(10\right)\end{array}$

[0047]
in the same manner as described for the determination of the coefficient k_{x }above. An example adaptation of coefficient k_{xy }is shown in FIG. 7B in which the horizontal axis represents the iteration number (i) of the calculation of coefficient k_{xy }and the vertical axis is the value of k_{xy}. A different coefficient k_{xy }can be adapted for each “corner” of the mirror angular range using this same procedure. This is particularly useful if the mirror 10, in its nominal at rest position, has a tilt with respect to the substrate, thus changing the effective separation d_{0}.

[0048]
The actual measurement of k
_{x }and k
_{y }can be interpreted as providing a more accurate value of the estimated gap d
_{0}. Because the torque and loop servo gain is proportional to (1/d
_{0})
^{2}, the adapted values of the coefficients k
_{x }and k
_{y }can be used to adapt the loop gain of the control system for a given mirror
10. In particular, the loop gain can be adapted as follows using the “measured values” rather than the design values:
$\begin{array}{cc}{k}_{\mathrm{gx}}={k}_{\mathrm{gx\_ref}}\xb7{\left(\frac{{k}_{\mathrm{x\_ref}}}{{k}_{x}}\right)}^{2}& \left(11\right)\\ {k}_{\mathrm{gy}}={k}_{\mathrm{gy\_ref}}\xb7{\left(\frac{{k}_{\mathrm{y\_ref}}}{{k}_{y}}\right)}^{2}& \left(12\right)\end{array}$

[0049]
in which k_{x} _{ — } _{ref }and k_{y} _{ — } _{ref }are the original design values for k_{x }and k_{y}, respectively, and k_{gx} _{ — } _{ref}, k_{gy} _{ — } _{ref}, k_{gx }and k_{gy }are the design gains and adapted loop gains, respectively, for positioning about the xaxis 18 and yaxis 26, respectively. A more consistent response is achieved for all mirrors in the array using this adaptation of the loop gain.

[0050]
In some applications, it is desirable to switch the various constants of derivative and integral feedback used in the mirror servo system 30. For example, one set of constants is optimized to achieve the minimum switching time during repositioning of the mirror 10. A different set of constants is preferred when the mirror 10 has reached its equilibrium position and in a tracking or lock mode. If parasitic capacitive coupling exists between the mirror 10 and its adjacent electrodes, it is often advantageous to select constants for the linear control module 54 (FIG. 3) so that the servo system interacts quickly to overcome any transients. For example, the integrator constant can be substantially increased.

[0051]
While the invention has been shown and described with reference to specific preferred embodiments, it should be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the spirit and scope of the invention as defined by the following claims. The method and apparatus of the present invention apply to both single mirror and multimirror systems. In addition, the method of the invention can be applied to any type of mirror system having a nonlinear response to an applied control signal.